Large Deviations for the Two-Dimensional Dean-Kawasaki Equation with Coulomb Interactions
Abstract
We establish global well-posedness and a small-noise large deviation principle for the two-dimensional Dean-Kawasaki equation with Coulomb interaction ∂tρε=Δρε−∇·(ρε(V*ρε))−√ε∇·(√(ρε)∘ξK(ε)), where V=λV(cos(αV)∇ G+sin(αV)J∇ G). Here λV≥0 is the interaction strength, αV is the interaction angle, G is the mean-zero Green function of −Δ on 𝕋2, J is rotation by π/2, and ξK is a finite-mode approximation of space-time white noise. For finite-entropy initial data of mass M satisfying λVmax{cos(αV),0}M<8π, the equation with the exact square-root coefficient is globally pathwise well posed at every finite Fourier cutoff in the class of stochastic kinetic solutions. Following Fehrman and Gess (arXiv:1910.11860), we prove a large deviation principle on L1((0,T)×𝕋2) under the scaling K(ε)→∞ and ε K(ε)4→0, with good rate function given by the quadratic control problem for the skeleton equation.
Related
- Well-Posedness of Dean–Kawasaki Equation with Singular Interactions
- Low-Temperature Large Deviations for the Two-Dimensional Wick-Ordered Cubic Wave Equation
- Uniform Large Deviations, Barrier Identity and the Néel--Arrhenius Law for the One-Dimensional Stochastic Landau--Lifshitz--Gilbert Equation
- A discontinuous Galerkin approximation of the Dean--Kawasaki equation
- Sharp Dispersive Estimates for the Schrödinger Equation with an Attractive Coulomb Potential
BibTeX
@misc{ji2026large,
title = {{Large Deviations for the Two-Dimensional Dean-Kawasaki Equation with Coulomb Interactions}},
author = {Xiaohao Ji},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23730},
eprint = {2609.23730},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23730},
}
RIS
TY - UNPB
TI - Large Deviations for the Two-Dimensional Dean-Kawasaki Equation with Coulomb Interactions
AU - Xiaohao Ji
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2609.23730
UR - https://arxiv.org/abs/2609.23730
AB - We establish global well-posedness and a small-noise large deviation principle for the two-dimensional Dean-Kawasaki equation with Coulomb interaction ∂_tρ^ε=Δρ^ε−∇·(ρ^ε(V*ρ^ε))−√ε∇·(√(ρ^ε)∘ξ^K(ε)), where V=λ_V(cos(α_V)∇ G+sin(α_V)J∇ G). Here λ_V≥0 is the interaction strength, α_V is the interaction angle, G is the mean-zero Green function of −Δ on 𝕋^2, J is rotation by π/2, and ξ^K is a finite-mode approximation of space-time white noise. For finite-entropy initial data of mass M satisfying λ_Vmax{cos(α_V),0}M<8π, the equation with the exact square-root coefficient is globally pathwise well posed at every finite Fourier cutoff in the class of stochastic kinetic solutions. Following Fehrman and Gess (arXiv:1910.11860), we prove a large deviation principle on L^1((0,T)×𝕋^2) under the scaling K(ε)→∞ and ε K(ε)^4→0, with good rate function given by the quadratic control problem for the skeleton equation.
ER -